absence of arbitrage for general admissible integrands (NA). From an economic viewpoint a local martingale measure Q, that gives zero measure to a non.
THE EXISTENCE OF ABSOLUTELY CONTINUOUS LOCAL MARTINGALE MEASURES
Freddy Delbaen Walter Schachermayer Department of Mathematics, Vrije Universiteit Brussel Institut f¨ ur Statistik, Universit¨ at Wien
Abstract. We investigate the existence of an absolutely continuous martingale measure. For continuous processes we show that the absence of arbitrage for general admissible integrands implies the existence of an absolutely continuous (not necessarily equivalent) local martingale measure. We also rephrase Radon-Nikodym theorems for predictable processes. 1.Introduction. In our paper Delbaen and Schachermayer (1994a) we showed that for locally bounded finite dimensional stochastic price processes S, the existence of an equivalent (local) martingale measure – sometimes called risk neutral measure – is equivalent to a property called No Free Lunch with Vanishing Risk (NFLVR). We also proved that if the set of (local) martingale measures contains more than one element, then necessarily, there are non equivalent absolutely continuous local martingale measures for the process S. We also gave an example, see Delbaen and Schachermayer (1994a) Example 7.7, of a process that does not admit an equivalent (local) martingale measure but for which there is a martingale measure that is absolutely continuous. The example moreover satisfies the weaker property of No Arbitrage with respect to general admissible integrands. We were therefore lead to the investigation of the relation between the two properties, the existence of an absolutely continuous martingale measure (ACMM) and the absence of arbitrage for general admissible integrands (NA). From an economic viewpoint a local martingale measure Q, that gives zero measure to a non negligible event, say F , poses some problems. The price of the contingent claim that pays one 1991 Mathematics Subject Classification. 90A09,60G44, 46N10,47N10,60H05,60G40. Key words and phrases. arbitrage, immediate arbitrage, martingale, local martingale, equivalent martingale measure, representing measure, risk neutral measure, stochastic integration, mathematical finance, predictable Radon Nikodym derivative. Part of this research was supported by the European Community Stimulation Plan for Economic Science contract Number SPES-CT91-0089. Typeset by AMS-TEX 1
unit of currency subject to the occurrence of the event F , is given by the probability Q[F ]. Since F is negligible for this probability, the price of the commodity becomes zero! In mosteconomic models preference relations are supposed to be strictly monotone and hence there would be an infinite demand for this commodity. At first sight the property (ACMM) therefore seems meaningless in the study of general equilibrium models. But as the present paper shows, for continuous processes it is a consequence of the absence of arbitrage (NA). We therefore think that the (ACMM) property has some interest also from the economic viewpoint. Throughout the paper all variables and processes are defined on a probability space (Ω,F,P). The space of all measurable functions, equipped with the topology of convergence in probability is denoted by L0 (Ω,F,P) or simply L0 (Ω) or L0 . If F ∈ F has non zero measure then the closed subspace of functions, vanishing on the complement F c of F is denoted by L0 (F ). The conditional probability with respect to a non negligible event F is denoted by PF and is defined ∩B] as PF [B] = P[F P[F ] . To simplify terminology we say that a probability Q that is absolutely continuous with respect to P is supported by the set F if Q is equivalent to PF , in particular we then have Q[F ] = 1. Indicator functions of sets F , etc. are denoted by 1F etc.. The probability space Ω is equipped with a filtration (Ft ) 0≤t T . (2) There is an S-integrable 1-admissible strategy K, ε > 0 and two stopping times T1 ≤ T2 such that T2 < ∞ on {T1 < ∞}, P[T2 < ∞] > 0, K = K1] T1 ,T2 ] and (K · S)T2 ≥ ε on the set {T2 < ∞}. Proof. Let S allow arbitrage and let H be a 1-admissible strategy that produces arbitrage, i.e., (H · S)∞ ≥ 0 with strict inequality on a set of strictly positive probability. We now distinguish two cases. Either the process H · S is never negative or the process H · S becomes negative with positive probability. In the first case let T = inf{t : (H · S)t > 0}. P∞ −n ˜ ˜ satisfies (1). H1[ T,T +θn [ . Then H Let (θn )∞ n=1 be a dense in ]0, 1[ and let H = n=1 2 We thank an anonymous referee for correcting a slip in a previous version of this paper at this point. In the second case we first look for ε > 0 such that P[inf t (H · S)t < −2ε] > 0. We then define T1 as the first time the process H · S goes under −2ε, i.e. T1 = inf{t | (H · S)t < −2ε}. On the set {T1 < ∞} we certainly have that the process H · S has to gain at least 2ε. Indeed at the end the process H · S is positive and therefore the time T2 = inf{t | t > T1 ,(H · S)t ≥ −ε} is finite on the set {T1 < ∞}. We now put K = H1] T1 ,T2 ] . The process K is 1-admissible since (K · S)t ≥ −1 + 2ε on the set {T1 < ∞}. Also (K · S)T2 ≥ ε on the set {T1 < ∞}. ¤ 8
3.2 Definition. We say that the semimartingale S admits immediate arbitrage at the stopping time T , where we suppose that P[T < ∞] > 0, if there is an S-integrable strategy H such that H = H1] T,∞]] , and (H · S)t > 0 for t > T . 3.3 Remark. (a) Let us explain why we use the term immediate arbitrage. Suppose S admits immediate arbitrage at T and that H is the strategy that realises this arbitrage opportunity. Clearly H · S ≥ 0 and (H · S)T +t > 0 for all t > 0 almost surely on {T < ∞}. Hence we can make an arbitrage almost surely immediately after the stopping time T has occured. (b) Lemma 3.1 shows that either we have an immediate arbitrage oppurtunity or we have a more conventional form of arbitrage. In the second alternative the strategy to follow is also quite easy. We wait until time T1 and then we start our strategy K. If the strategy starts at all (i.e. if T1 < ∞) then we are sure to collect at least the amount ε in a finite time. It is clear that such a form of arbitrage is precisely what one wants to avoid in economic models. The immediate arbitrage seems, at first sight, to be some mathematical pathology that can never occur. However the concept of immediate arbitrage can occur as the following example shows. In model building one therefore cannot neglect the phenomenon. 3.4 Example. Take the one-dimensional Brownian motion W = (Wt )t∈[0,1] with its usual √ filtration. For the price process S we take St = Mt + At = Wt + t which satisfies the √ . We will show that such a situation leads to “immediate” differential equation dSt = dWt + 2dt t 1 . With this choice the integral on the drift-term arbitrage at time T = 0. Take Ht = √t(ln t)2 Rt √ = 1 (ln(t−1 ))−1 is convergent. Hu 2du 2 0 u Rt As for the martingale part, the random variable 0 Hu dWu has variance Rt 1 du which is of the order ln(t−1 )−3 . The iterated logarithm law implies that for 0 u(ln u)4 p t = t(ω) small enough |(H · W )t (ω)| ≤ C (ln(t−1 ))−3 ln ln((ln(t−1 ))3 ) ≤ C 0 (ln(t−1 ))−5/4 . It follows that, for t small enough, we necessarily have that (H · S)t (ω) > 0. We now define the stopping time T as T = inf{t > 0 | (H · S)t = 0} and, for n > 0, Tn = T ∧ n−1 . Clearly T (H · S) P∞≥ 0 andTnP[(H · S)Tn > 0] tends to 1 as n tends to infinity. By considering the integrand L = n=1 αn H for a sequence αn > 0 tending to zero sufficiently fast, we can even obtain that (L · S)t is almost surely strictly positive for each t > 0. We now give some more motivation why such a form of arbitrage is called immediate arbitrage. In the preceding example, for each stopping time T > 0 the process RS − S T admits 1 an equivalent martingale measure Q(T ) given by the density fT = exp(− 12 T √1u dWu − R1 1/8 T u1 du). We can check this by means of the Girsanow-Maruyama formula or we can checkit even more directly via Itˆ o’s rule. This statement shows that if one wants to make an arbitrage profit one has to be very quick since a profit has to be the result of an action taken before time T . Let us also note that the process S also satisfies the (NA) property for simple integrands. As is well known it suffices to consider integrands of the form f 1] T0 ,T1 ] where f is FT0 -measurable (see Delbaen and Schachermayer (1994e)). Let us show that such an integrand does not allow an arbitrage. Take T0 ≤ T1 two stopping times. We distinguish between P[T0 > 0] = 1 and T0 = 0. (The 0 − 1 law for F0 (Blumenthal’s theorem) shows that one of the two holds). 9
If T0 > 0, P a.s. then the result follows immediately from the existence of the martingale measure Q(T0 ) for the process S − S T0 . If T0 = 0 we have to prove that ST1 ≥ 0 (or ST1 ≤ 0) implies that ST1 = 0 a.s.. We concentrate on the first case and assume to the contrary that ST1 ≥ 0 and P{ST1 > 0} > 0. Note that it follows from the law of the iterated logarithm that inf{t|St < 0} = 0 almost surely, hence the stopping times Tε = inf{t > ε|St < −ε} tend to zero a.s. as ε tends to zero. Let ε > 0 be small enough such that {Tε < T1 } has positive measure to arrive at a contradiction: 0 > EQ(Tε ) [STε 1(Tε 0 a.s.. T = inf t | 0
(2) The [0, ∞]-valued increasing process does not jump to ∞.
Rt 0
h0u dhM ,M iu hu is continuous; in particular it
10
Proof. The existence of the process h follows from the preceding theorem. The stopping time T is well defined. The first claim on the stopping time T follows from the second, so we limit the proof to the second statement. We will prove that the set Z
T +ε
F = {T < ∞} ∩ {
h0t dhM, M it ht = ∞ ∀ε > 0}
T
has zero measure. Clearly F is, by right continuity of the filtration, an element of the σ-algebra FT . As the process hM, M it is continuous, assertion (2) will follow from the fact that P[F ] = 0. Suppose now to the contrary that F has strictly positive measure. We then look at the process 1F (S − S T ), adapted to the filtration (FT +t )t≥0 and we replace the probability P by PF . With this notation the theorem is reduced to the case T = 0. This case is treated in the following theorem. It is clear that this will end the proof. ¤ 3.7 Immediate Arbitrage Theorem. Suppose the d-dimensional continuous semi- martingale S has a Doob-Meyer decomposition given by dSt = dMt + dhM, M it ht where h is a d-dimensional predictable process. Suppose that a.s. Z
ε
(1)
h0t dhM, M it ht = ∞
∀ ε > 0.
0
Then for all ε > 0, there is an S-integrable strategy H such that H = H1[ 0,ε]] , H · S ≥ 0 and P[(H · S)t > 0] = 1, for each t > 0. In other words S admits immediate arbitrage at ¿time T = 0. Proof. The proof of the theorem is based on the following lemma 3.8 Lemma. If (1) holds almost surely then for any a, ε, η > 0 we can find 0 < δ < ε/2 and an a-admissible integrand H with H =H1 R ε 0 ] δ,ε]] R ε 0 |Hs dA|s + δ Hs dhM, M is Hs < 2 + a δ P [(H · S)ε ≥ 1] ≥ 1 − η. Proof of the Lemma. Fix a, ε, η > 0 and let R ≥ max{ η8 almost surely, we have that ·Z lim
δ↓0,K↑∞
ε
P δ
1{|h|≤K} h0t
¡ 1+a ¢2 a
, (1 + a)2 }. Since (1) is satisfied ¸
dhM, M it ht ≥ R = 1.
Hence we can find a K > 0 and a 0 < δ < ε/2 such that Z ∞> δ
ε
1{|h|≤K} h0t dhM, M it ht ≥ R 11
on a Fε -measurable set Λ with P [Λ] ≥ 1 − η2 . Let Z t 1{|h|≤K} h0t dhM, M it ht ≥ R} ∧ ε T = inf{t > 0 | δ
and let H =
1+a R
h1] δ,T ] 1{|h|≤K} . Then Z ε (1 + a)2 Hs0 dhM, M is Hs ≤ R 0 Z
and
ε
|Hs dA|s ≤ (1 + a) a.s.. 0
Therefore H is S-integrable. Moreover, (H · A)ε = 1 + a on Λ. Rε 2 we obtain from Doob’s inequality together Since kH · M k22 = E[ 0 Hs0 dhM, M is Hs ] ≤ (1+a) R 2 with Tchebychew’s inequality (both in their L –version) µ
∗
P [(H · M ) ≥ a] ≤ 4
(2)
1+a a
¶2
1 η ≤ R 2
We now localize H to be a-admissible. Let T2 = inf {t > 0 | (H · M )t < −a} ∧ T. Then T2 = T on {(H · M )∗ ≤ a} and from (2) we obtain ¤ £ P (H 1[ 0,T2 ] · S)ε ≥ 1 ≥ P [{(H · A)ε ≥ 1 + a} ∩ {(H · M )∗ < a}] ≥ P [Λ] − P [(H · M )∗ ≥ a] ≥ 1 − η which proves the lemma.
¤
Proof of the Immediate Arbitrage Theorem. Assume that (1) is valid for almost every ω ∈ Ω. We will now construct an integrand which realizes immediate arbitrage. Let ε0 > 0 be such that ε0 ≤ min (ε, 12 ). By lemma 3.8 we can find a stricly decreasing sequence of positive numbers (εn )n≥0 with limn→∞ εn → 0 and integrands Hn = Hn 1] εn+1 ,εn ] such that Hn is 4−n – R εn R εn |(Hn )0s dAs | + εn+1 (Hn )0s dhM, M is (Hn )s < 3/2n and P [(Hn · S)εn ≥ 2−n ] ≥ admissible, εn+1 ˆ = P∞ Hn . Then H ˆ is S-integrable. Define 1 − 2−n . Let H n=1 n o ˆ · S)t = 0 . T (ω) = inf t > 0 | (H We claim that T (ω) > 0 for almost every ω ∈ Ω. Since P [(Hn · S)εn < 2−n ] ≤ 2−n , we obtain from the Borel–Cantelli Lemma that for almost every ω ∈ Ω there is a N (ω) ∈ N with (Hn · S)εn (ω) > 2−n for all n > N (ω). If n > N (ω) and εn+1 < t ≤ εn then ˆ · S)t (ω) = (H
∞ X
1 (Hk · S)εk (ω) + (Hn · S)t (ω) ≥ n+1 | {z } 2 k>n ≥−2−(n+1) {z } | ≥2−n
12
and we have verified the claim. Hence h³ ´ i ˆ 1[ 0,T ] · S > 0 = 1 . lim P H t→0
t
Finally let H=
∞ X
ˆ ] 0,T ∧ε [ 2−n H1 n
n=1
to find an S-integrable predictable process supported by [0, ε] such that (H · S)t > 0 for each t > 0. ¤ 4. The existence of an absolutely continuous martingale measure. We start this section with the investigation of the support of an absolutely continuous risk neutral measure. The theory is based on the analysis of the density given by a Girsanow– Maruyama transformation. If dSt = dMt + dhM, M it ht defines the Doob-Meyer decomposition of a continuous semimartingale, where h is a d-dimensional predictable process and where M is a d-dimensional continuous local martingale, then the Girsanov–Maruyama transformation is, at Rt Rt least formally, given by the local martingale Lt = exp( 0 −h0u dMu − 1/2 0 h0u dhM, M iu hu ), L0 = 1. Formally one can verify that LS is a local martingale. However, things are not so easy. First of all, there is no guarantee that the process h is M -integrable, so L need not be defined. Second, even if L is defined, it may only be a local martingale and not a uniformly integrable martingale. The examples in Schachermayaer (1993) and in Delbaen and Schachermayer (1994b) show that even when an equivalent risk neutral measure exists, the local martingale L need not to be uniformly integrable. In other words a risk neutral measure need not be given by L. Third, in case the two previous points are fulfilled, the density L∞ need not be different from zero a.s.. What can we save in our setting? In any case, theorem 3.6 shows that in the case when S satisfies the No Arbitrage property for general admissible integrands, the process h satisfies the properties (1) Z t h0 dhM ,M i h = ∞} > 0 a.s.. T = inf{t | Rt
0
(2) The [0, ∞]-valued proces 0 h0u dhM ,M iu hu is continuous; in particular it does not jump to ∞. In this case the stochastic integrals h · M and h · S can be defined on the interval [ 0, T [ and at time T we have that LT can be defined as the left limit. The theory of continuous martingales (Revuz and Yor (1991)) shows that (Z ) T
{LT = 0} =
h0t dhM, M it ht = ∞ .
0
If after time T , i.e. for t > T , we put Lt = 0 the process L is well defined, it is a continuous local martingale, satisfies dLt = −Lt h0t dMt and LS is a local martingale. The process X = L1 − 1 13
is also defined on the interval [ 0, T [ and on the set {LT = 0} its left limit equals infinity. The crucial observation is now that on the interval [ 0, T [ , we have that dXt = L1t h0t dSt . This follows simply by plugging in Itˆ o’s formula (compare Delbean and Schachermayer (1994c)). For each ε > 0 let τ ε be the stopping time defined by τ ε = inf{t | Lt ≤ ε}. Because the ε process X is always larger than −1, the stopped processes X τ are outcomes of admissible integrands. If Q is an absolutely continuous probability measure© such that S ª becomes a loτε | ε > 0 is bounded in cal martingale than, by theorem 1.3 we have that the set H = X∞ L0 ({ dQ dP > 0}). But it is clear that on the set {LT = 0}, the set H is unbounded. As a consequence we obtain the following 4.1 Lemma. If the continuous semimartingale S satisfies the No Arbitrage condition with respect to general admissible integrands and if Q is an absolutely continuous local martingale measure for S, then { dQ dP > 0} ⊂ {LT > 0}. In order to prove the existence of an absolutely continuous local martingale measure Q we therefore should restrict ourselves to measures supported by F = {LT > 0}. Note that the No Arbitrage condition implies that P[F ] > 0. Indeed, suppose that P[F ] = 0 and let 1 U = inf{t : Lt ≤ }. 2 1 We than have that P[U < ∞] = 1, LU ≡ 2 and therefore XU ≡ 1. Hence H = L1 h0 1[ 0,U ] is a 1-admissible integrand such that (H · S)∞ ≡ XU ≡ 1, a contradiction to (NA). So we will look at the process S under the conditional probability measure PF . Our strategy will be to verify that S satisfies the property (NFLVR) with respect to PF which will imply the existence of a local martingale measure Q for S which is equivalent to PF and therefore absolutely continuous with respect to P . But there are difficulties: Under the measure PF the Doob-Meyer decomposition will change, there will be more admissible integrands and the verification of the No Free Lunch Property with Vanishing Risk for general admissible integrands (under PF !!) is by no means trivial. We are now ready to reformulate the main theorem stated in the introduction in a more precise way and to abord the proof: 4.2 Main Theorem. If the continuous semimartingale S satisfies the No Arbitrage property with respect to general admissible integrands, then with the notation introduced above, it satisfies the No Free Lunch Property with Vanishing Risk with respect to PF . As a consequence there is an absolutely continuous local martingale measure that is equivalent to PF , i.e. it is precisely supported by the set F . The proof of the theorem still needs some auxiliary steps which will be stated below. We first deal with the problem of the usual hypotheses under the measure PF . The σ-algebras F˜t of the PF -augmented filtration are obtained from Ft by adding all PF null sets. It is easily seen that the new filtration is still right continuous and satisfies the usual hypotheses for the new measure PF . The following technical results are proved in Delbaen and Schachermayer (1994d). 14
4.3 Proposition. If τ˜ is a stopping time with respect to the filtration (F˜t )t≥0 then there is a stopping time τ with respect to the filtration (Ft )t≥0 such that PF a.s. we have τ˜ = τ . If τ˜ is finite or bounded then τ may be chosen to be finite or bounded. ˜ is a predictable process with respect to the filtration (F˜t )t≥0 then there is 4.4 Proposition. If H ˜ = H. a predictable process H with respect to the filtration (Ft )t≥0 , such that PF a.s. we have H ˜ This settles the problem of the usual hypotheses. Each time we need a F-predictable process, we can without danger, replace it by a predictable process for F. Without further notice we will do this. ˜ PF ). This is well known, The process S is a semimartingale with respect to the system (F, see Protter (1990). R∞ Remark also that for PF we have that 0 h0u dhM, M iu hu < ∞ a.s.. We will need this later on. As a first step we will decompose S into a sum of a PF -local martingale and a predictable process of finite variation. Because PF is only absolutely continuous with respect to P we need an extension of the Girsanov-Maruyama formula for this case. The generalisation was given by Lenglart (1977). We need the cadlag martingale U defined as ¸ · 1F | Ft Ut = E P[F ] Note that U is not necessarily continuous, as we only assumed that S is continuous and not that each Ft -martingale is continuous. Together with the process U we need the stopping time ν = inf{t | Ut = 0} = inf{t > 0 | Ut− = 0} (see Dellacherie and Meyer (1980) for this equality). 4.5 Lemma. ν = T P-almost surely. Proof of the lemma. We first show that for an arbitrary stopping time σ we have that Lσ > 0 on the set {Uσ > 0}. Let A be a set in Fσ such that P[A ∩ {Uσ > 0}] > 0. This already implies that P[A ∩ F ] > 0. Indeed we have that E[1A 1F | Fσ ] = P[F ]1A Uσ and hence we necessarily have that P[A ∩ F ] > 0. The following chain of equalities is almost trivial Z Z Z Z Z Lσ = Lσ 1{Uσ >0} ≥ P[F ] Lσ Uσ = Lσ 1F = Lσ . A∩{Uσ >0}
A
A
A
A∩F
The last term is strictly positive sinceR Lσ > 0 on F . This proves that for each set A such that P[A ∩ {Uσ > 0}] > 0 we must have A∩{Uσ >0} Lσ > 0. This implies that Lσ > 0 on the set {Uσ > 0}, hence ν ≤ T . The converse inequality is less trivial and requires the use of the (NA) property of S. We proceed in the same way. Take G ∈ Fσ such that G ⊂ {Lσ > 0} and P[G] > 0. Suppose that 15
Uσ = 0 on G. We will show that this leads to a contradiction. If Uσ = 0 on G then clearly G ∩ F = ∅. But on F c we have that Lt tends to zero and hence L1t tends to ∞. We know that 1 Lt − 1 can be obtained as a stochastic integral with respect to S. We take the stopping time µ = ∞ on Gc and equal to inf{t | Lt ≤ 12 Lσ } on the set G. The outcome µ 1G =
1 1 − Lµ Lσ
¶ Lσ 1G
is the result of a 1-admissible strategy and Rclearly produces arbitrage. We may therefore suppose that P[G ∩ F ] > 0 and hence we also have G Uσ > 0. Again this suffices to show that Uσ > 0 on the set {Lσ > 0} and again implies that T ≤ ν. The proof of the lemma is complete now. ¤ Proof of the Main Theorem. We now calculate the decomposition of the continuous semimartindecomposition of S under P then, under gale S under PF . If S = M + A is the Doob-Meyer R ˜ + A˜ where A˜t = At + dhM,U is , see Lenglart (1977). This integral exists PF we write S = M Us for the measure PF since on F the process U is bounded away from 0. A more explicit formula for A˜ can be found if we use the structure of hM, U i. We thereto use the Kunita-Watanabe decomposition of the L2 martingale U with respect to the martingale M . This is done in the following way (see Jacod (1979)). The spaceRof all L2 martingales of the form α · M is a stable space and in fact we have k(α · M )∞ k2 = E[ α0 dhM, M i α]. The orthogonal projection of U∞ on this space is given by (β · M )∞ for some predictable process β, where of course Z E[ β 0 dhM, M i β] < ∞. In this notation we may write: It follows that also written as
R
dhM, U i = dhM, M iβ. β 0 dhM, M i β < ∞ a.s. for the measure PF and the measure dA˜ can be dA˜ = dhM, M i(ht +
βt ) = dhM, M i kt . Ut
Here we have put k = h + Uβ to simplify notation. To prove the N F LV R property for S under PF we use the criterion of theorem 1.3 above. Step 1: the set of 1-admissible integrands for PF is bounded in L0 (F ) From the properties of β and h we deduce that for the measure PF , the integral Z
∞
kt0 dhM, M ikt < ∞ PF a.s.
0
˜ is now defined as The PF local martingale L µ Z t ¶ Z 1 t 0 0 ˜ ˜ ku dMu − k dhM, M iu ku Lt = exp − 2 0 u 0 16
It follows that ˜ ∞ > 0 PF a.s. L ˜ is a PF local martingale and therefore the set K˜1 constructed It is chosen in such a way that LS with the 1-admissible, with respect to PF !!!, integrands, is bounded in L0 (PF ). In particular this also excludes the possibility of immediate arbitrage for S with respect to PF . Step 2: S satisfies (NA) with respect to PF (and with respect to general admissible integrands). Since by step 1 immediate arbitrage is excluded the violation of the (NA) property would, by lemma 3.1, give us a predictable integrand H such that for PF the integrand is of finite support, is S-integrable and 1-admissible. When the support of H is contained in ] σ1 , σ2 ] it gives an outcome at least ε on the set {σ1 < ∞}. All this, of course, with respect to PF . The rest of the proof is devoted to the transformation of this phenomenon to a situation valid for P. Without loss of generality we may suppose that for the measure P we have σ1 ≤ σ2 ≤ T , we replace e.g. the stopping time σ2 by max(σ1 , σ2 ) and then we replace σ1 and σ2 by respectively min(T, σ1 ) and min(T, σ2 ). All these substitutions have no effect when seen under the measure PF . Since PF [{σ1 < σ2 < ∞}] > 0, we certainly have that P[{σ1 < σ2 < T }] > 0. Roughly speaking we will now use the strategy H to construct arbitrage on the set F and we use the process L1 to construct a sure win on the set F c , as on the interval [ 0, T [ , the process 1 L − 1 equals K · S for a well chosen integrand K. When we add the two integrands, H and K, we should obtain an integrand that gives arbitrage on Ω with respect to P and this will provide the desired contradiction. © ª Let the sequence of stopping times τn be defined as τn = inf t | Lt ≤ n1 . We have that τn ↑ T for P and τn ↑ ∞ for the measure PF . Since we have that Lτn > 0 a.s. we also have that Uτn > 0 a.s.. It follows that on the σ-algebra Fτn the two measures, P and PF are equivalent. We can therefore conclude that for each n the integrand H1[ 0,τn ] as well as the integrand K1[ 0,τn ] is S integrable and 1-admissible for P. The last integrand still has to be renormalised. In fact on the set F itself, the lower bound −1 for the process K · S is too low since it will be compensated at most by ε. We therefore transform K in such a way that it will stay above ε/2 but will nevertheless give outcomes that are very big on the set F c . Let us define ˜ = K1{σ
